Overview

The goal of the flipr package is to provide a flexible framework for making inference via permutation. The idea is to promote the permutation framework as an incredibly well-suited tool for inference on complex data. You supply your data, as complex as it might be, in the form of lists in which each entry stores one data point in a representation that suits you and flipr takes care of the permutation magic and provides you with either point estimates or confidence regions or $p$-value of hypothesis tests. Permutation tests are especially appealing because they are exact no matter how small or big your sample sizes are. You can also use the so-called non-parametric combination approach in this setting to combine several statistics to better target the alternative hypothesis you are testing against. Asymptotic consistency is also guaranteed under mild conditions on the statistic you use. The flipr package provides a flexible permutation framework for making inference such as point estimation, confidence intervals or hypothesis testing, on any kind of data, be it univariate, multivariate, or more complex such as network-valued data, topological data, functional data or density-valued data.

Installation

You can install the latest stable version of flipr on CRAN with:

install.packages("flipr")

Or you can install the development version from GitHub with:

# install.packages("remotes")
remotes::install_github("LMJL-Alea/flipr")

Example

library(flipr)

We hereby use the very simple t-test for comparing the means of two univariate samples to show how easy it is to carry out a permutation test with flipr.

Data generation

Let us first generate two samples of size $15$ governed by Gaussian distributions with equal variance but different means:

set.seed(123)
n <- 15
x <- rnorm(n = n, mean = 0, sd = 1)
y <- rnorm(n = n, mean = 1, sd = 1)

Given the data we simulated, the parameter of interest here is the difference between the means of the distributions, say $\delta = \mu_y - \mu_x$.

Make the two samples exchangeable under $H_0$

In the context of null hypothesis testing, we consider the null hypothesis $H_0: \mu_y - \mu_x = \delta$. We can use a permutation scheme to approach the $p$-value if the two samples are exchangeable under $H_0$. This means that we need to transform for example the second sample to make it exchangeable with the first sample under $H_0$. In this simple example, this can be achieved as follows. Let $X_1, \dots, X_{n_x} \sim \mathcal{N}(\mu_x, 1)$ and $Y_1, \dots, Y_{n_y} \sim \mathcal{N}(\mu_y, 1)$. We can then transform the second sample as $Y_i \longleftarrow Y_i - \delta$.

We can define a proper function to do this, termed the null specification function, which takes two input arguments:

• y which is a list storing the data points in the second sample;
• parameters which is a numeric vector of values for the parameters under investigation (here only $\delta$ and thus parameters is of length $1$ with parameters[1] = delta).

In our simple example, it boils down to:

null_spec <- function(y, parameters) {
  purrr::map(y, ~ .x - parameters[1])
}

Choose suitable test statistics

Next, we need to decide which test statistic(s) we are going to use for performing the test. Here, we are only interested in one parameter, namely the mean difference $\delta$. Since the two samples share the same variance, we can use for example the $t$-statistic with a pooled estimate of the common variance.

This statistic can be easily computed using stats::t.test(x, y, var.equal = TRUE)$statistic. However, we want to extend its evaluation to any permuted version of the data. Test statistic functions compatible with flipr should have at least two mandatory input arguments:

• data which is either a concatenated list of size $n_x + n_y$ regrouping the data points of both samples or a distance matrix of size $(n_x + n_y) \times (n_x + n_y)$ stored as an object of class dist.
• indices1 which is an integer vector of size $n_x$ storing the indices of the data points belonging to the first sample in the current permuted version of the data.

Some test statistics are already implemented in flipr and ready to use. User-defined test statistics can be used as well, with the use of the helper function use_stat(nsamples = 2, stat_name = ). This function creates and saves an .R file in the R/ folder of the current working directory and populates it with the following template:

#' Test Statistic for the Two-Sample Problem
#'
#' This function computes the test statistic...
#'
#' @param data A list storing the concatenation of the two samples from which
#'   the user wants to make inference. Alternatively, a distance matrix stored
#'   in an object of class \code{\link[stats]{dist}} of pairwise distances
#'   between data points.
#' @param indices1 An integer vector that contains the indices of the data
#'   points belong to the first sample in the current permuted version of the
#'   data.
#'
#' @return A numeric value evaluating the desired test statistic.
#' @export
#'
#' @examples
#' # TO BE DONE BY THE DEVELOPER OF THE PACKAGE
stat_{{{name}}} <- function(data, indices1) {
  n <- if (inherits(data, "dist"))
    attr(data, "Size")
  else if (inherits(data, "list"))
    length(data)
  else
    stop("The `data` input should be of class either list or dist.")

  indices2 <- seq_len(n)[-indices1]

  x <- data[indices1]
  y <- data[indices2]

  # Here comes the code that computes the desired test
  # statistic from input samples stored in lists x and y

}

For instance, a flipr-compatible version of the $t$-statistic with pooled variance will look like:

my_t_stat <- function(data, indices1) {
  n <- if (inherits(data, "dist"))
    attr(data, "Size")
  else if (inherits(data, "list"))
    length(data)
  else
    stop("The `data` input should be of class either list or dist.")

  indices2 <- seq_len(n)[-indices1]

  x <- data[indices1]
  y <- data[indices2]
  
  # Here comes the code that computes the desired test
  # statistic from input samples stored in lists x and y
  x <- unlist(x)
  y <- unlist(y)
  
  stats::t.test(x, y, var.equal = TRUE)$statistic
}

Here, we are only going to use the $t$-statistic for this example, but we might be willing to use more than one statistic for a parameter or we might have several parameters under investigation, each one of them requiring a different test statistic. We therefore group all the test statistics that we need into a single list:

stat_functions <- list(my_t_stat)

Assign test statistics to parameters

Finally we need to define a named list that tells flipr which test statistics among the ones declared in the stat_functions list should be used for each parameter under investigation. This is used to determine bounds on each parameter for the plausibility function. This list, often termed stat_assignments, should therefore have as many elements as there are parameters under investigation. Each element should be named after a parameter under investigation and should list the indices corresponding to the test statistics that should be used for that parameter in stat_functions. In our example, it boils down to:

stat_assignments <- list(delta = 1)

Use the plausibility function

Now we can instantiate a plausibility function as follows:

pf <- PlausibilityFunction$new(
  null_spec = null_spec,
  stat_functions = stat_functions,
  stat_assignments = stat_assignments,
  x, y
)
#> ! Setting the seed for sampling permutations is mandatory for obtaining a continuous p-value function. Using `seed = 1234`.

Now, assume we want to test the following hypotheses:

$$ H_0: \delta = 0 \quad \mbox{v.s.} \quad H_1: \delta \ne 0. $$

We use the $get_value() method for this purpose, which essentially evaluates the permutation $p$-value of a two-sided test by default:

pf$get_value(0)
#> [1] 0.1078921

We can compare the resulting $p$-value with the one obtained using the more classic parametric test:

t.test(x, y, var.equal = TRUE)$p.value
#> [1] 0.1030946

The permutation $p$-value does not quite match the parametric one. This is because of two reasons:

1. The resolution of a permutation $p$-value is of the order of $1/(B+1)$, where $B$ is the number of sampled permutations. By default, the plausibility function is instantiated with $B = 1000$:

pf$nperms
#> [1] 1000

2. We randomly sample $B$ permutations out of the $\binom{n_x+n_y}{n_x}$ possible permutations and therefore introduce extra variability in the $p$-value.

If we were to ask for more permutations, say $B = 1,000,000$, we would be much closer to the parametric $p$-value:

pf$set_nperms(1000000)
pf$get_value(0)
#> [1] 0.1029879